Miguel Piñar

Affiliation: Universidad de Granada, Spain.

Stefano De Marchi

Affiliation: Department of Medicine, University of Padova, Italy.

From Radial Basis Functions to VS(D,R) kernels

Radial Basis Functions (RBF) have become quite a popular tool in mathematics and applied sciences, for their flexibility, allowing us to work with multidimensional data, without requiring any special structure of the data to be approximated. Usually, RBF depends on a shape parameter that needs to be tuned or optimised. Variably Scaled Kernels, or their counterparts for discontinuous functions, called Variably Scaled Discontinuous Kernels or the Variably Scaled Rational Kernels, are known alternatives that use a scaling function that allows for better mimicking the data. In this talk, after a review of the most important properties of RBF approximations, we present VS(D, R) kernels and some of their applications.

Gerlind Plonka-Hoch

Affiliation: Institute for Numerical and Applied Mathematics, University of Göttingen, Germany.

EXPOWER: Methods for reconstruction of parametric exponential models

We consider the parametric exponential model

\[ f(t) = \sum_{j=1}^{M} \gamma_{j} \, \mathrm{e}^{\phi_{j}t} = \sum_{j=1}^{M} \gamma_{j} \, z_{j}^{t}, \]

where \(M \in \mathbb{N}\), \(\gamma_{j} \in \mathbb{C}\setminus\{0\}\), and \(z_{j} = \mathrm{e}^{\phi_{j}} \in \mathbb{C}\setminus\{0\}\) with \(\phi_{j} \in \mathbb{C}\) are pairwise distinct. The recovery of such exponential sums from a finite set of possibly corrupted signal samples plays an important role in many signal processing applications, as in phase retrieval, signal approximation, sparse deconvolution in nondestructive testing, model reduction in system theory, direction of arrival estimation, exponential data fitting, or reconstruction of signals with finite rate of innovation. Models of the above form are closely related to other parametric function models, as e.g. \(f(t) = \sum_{j=1}^{M} \gamma_{j} \phi(t-T_{j})\) with a given basis function \(\phi\) and arbitrary \(T_{j} \in \mathbb{R}\), or \(f(t) = \sum_{j=1}^{M} \gamma_{j} \cos(2\pi a_{j} t + b_{j})\).

There is however also a close relation to rational approximation models of the form \(f(z) = \sum_{j=1}^{N} \frac{a_{j}}{t - b_{j}}\).

All parameters of these models can be recovered using the Prony method or its generalizations. However, to improve the stability of the reconstruction, we want to exploit the strong connections between the algebraic methods and other approaches from rational approximation, structured matrix theory and optimization. This talk gives a survey on different numerical approaches to the recovery problem for exponential models.

Bernhard Beckermann

Affiliation: Laboratoire Painlevé, Université de Lille, France.

Rational approximation of Markov functions

The aim of this talk is to give new explicit upper bounds for the relative error of best rational approximents of Markov functions (that is, the Cauchy transform of some measure \(\mu\) with real support). In the special case of the Markov function \(f(z)=1/\sqrt{z}\), explicit formulas and bounds have been given already by Zolotarev more then 100 years ago, but are still subject of interest, see for instance a recent paper of Braess and Hackbusch. We will show that, within the class of measures satisfying the Szegő condition, it is known that, asymptotically, the worst case measure with support in \([a,b]\) is the equilibrium measure. For general measures on \([a,b]\) such like discrete measures, we show that there is a worst case measure, and thus we get new sharp upper bounds depending only on the capacity of the underlying condenser, and the degree of the rational function.

Stefan Güttel

Affiliation: University of Manchester, UK.

Shared-denominator rational approximation

Several problems in scientific computing can be related to approximating a family of parameter-dependent scalar functions, say \(f^\tau(z)\), by rational functions \(r^\tau(z)\) that all share the same denominator. Of particular interest are the resolvent function \(f^\tau(z) = (z-\tau)^{-1}\) (arising in frequency domain model order reduction), the exponential family \(f^\tau(z) = \exp(\tau z)\) (time domain), and the so-called \(\varphi\)-functions (exponential integrators). I will discuss some of the applications and report on recent work on the theory and computation of shared-denominator rational approximants.